Discover the Ellipse From planetary paths to classroom graphs—ellipses unveiled.

String-Pin Definition

String-pin construction of an ellipse

Constant-Sum Property

An ellipse is the locus of all points \(P\) such that \(PF_{1}+PF_{2}\) remains constant.

Key Points:

  • Constant string length gives \(PF_{1}+PF_{2}=2a\).
  • Two foci are needed; one focus would form a circle.

Naming the Parts

Ellipse diagram

Standard ellipse with axes, centre, vertices, and foci.

Parts of an Ellipse

An ellipse has two perpendicular symmetry lines: the major axis and the minor axis.

The major axis is the longest chord; its endpoints are called vertices.

The minor axis is the shortest chord and meets the major axis at the centre.

Two fixed points on the major axis, equidistant from the centre, are the foci.

Key Points:

  • Major axis: longest chord, length \(2a\).
  • Minor axis: shortest chord, length \(2b\).
  • Vertices: endpoints of major axis \((\pm a,0)\).
  • Centre: midpoint of both axes \((0,0)\).
  • Foci: points \((\pm c,0)\) where \(c^{2}=a^{2}-b^{2}\).

Parameters a, b, c

Ellipse with parameters a, b, c

Semi-axes and focal distance

The ellipse is governed by three parameters: semi-major axis \(a\), semi-minor axis \(b\) and focal distance \(c\).

These values define its size, flatness and the position of its two foci.

Key Points:

  • Relationship: \(b^{2}=a^{2}-c^{2}\).
  • As \(c\) approaches \(a\), \(b\) decreases and the ellipse stretches.
  • If \(c=0\), then \(b=a\); the curve becomes a circle.

Standard Forms

https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/r4MwzUv1GPCLUsbCCT6cIeFeGXngltRGvr1MJTCC.png

Ellipse with horizontal and vertical axes

Ellipse: choose the right standard equation

Identify the major axis, then write the matching standard equation.

Key Points:

  • Horizontal major axis: \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \)
  • Vertical major axis: \( \frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1 \)
  • For both, \( b^{2} = a^{2} - c^{2} \)

e vs b/a Curve

Curve of b/a vs e

Curve of \( \frac{b}{a} = \sqrt{1-e^{2}} \)

Eccentricity vs Semi-minor Ratio

Graph displays the relationship \( \frac{b}{a} = \sqrt{1-e^{2}} \) between eccentricity and semi-minor ratio.

As \(e\) rises, \(b/a\) falls, flattening the ellipse.

Key Points:

  • \(e = 0\): \(b/a = 1\) — circle
  • \(e = 0.6\): \(b/a \approx 0.8\) — moderate flattening
  • \(e \to 1\): \(b/a \to 0\) — almost a line

b² = a²(1 – e²)

1
\[c = ae\]

Definition of eccentricity: focus distance is e times semi-major axis.

2
\[b^{2} = a^{2} - c^{2}\]

Ellipse axes satisfy a right-triangle relation at any vertex.

3
\[b^{2} = a^{2} - a^{2}e^{2} = a^{2}(1 - e^{2})\]

Substitute \(c = ae\) and simplify to link b, a, and e.

Key Insight:

Greater eccentricity shrinks \(1 - e^{2}\), so the semi-minor axis shortens while a stays fixed.

Multiple Choice Question

Question

For an ellipse with \(a = 5\) and \(c = 3\), find \(b\).

1
4
2
\( \sqrt{16} \)
3
3
4
\( \sqrt{34} \)

Hint:

Use \(b = \sqrt{a^{2} - c^{2}}\).

Ellipse Essentials

Locus definition: sum of distances to two foci remains constant.

Key parts: centre, foci, vertices, major axis, minor axis.

Parameters satisfy \(b^{2}=a^{2}-c^{2}\).

Standard forms: horizontal \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \), vertical \( \frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1 \).

Eccentricity \(e=\frac{c}{a}\) shows stretch; \(0<e<1\).

Thank You!

We hope you found this lesson informative and engaging.